Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions. Use this tag along with the tags (probability), (probability-theory) or (statistics).

Any probability distribution, including beta, binomial, chi, Erlang, gamma, geometric, lognormal, negative binomial, normal (Gaussian), Pareto, Poisson, Student's t, uniform, Wald, Weibull, zeta, and Zipf.

28080 questions
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Distribution of transformed poisson distribution

Let X follows Poisson distribution with parameter $\lambda\gt0$ and $Y=aX$, where $a\gt0$ is a constant. Q. What will be the $PMF$ of $Y$? Since by using the M.G.F we have $M_Y(t)=M_X(at)=(e^{\lambda(e^{at}-1)})$. What next I should do to derive its…
user2586518
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Sum of folded normal distributions

Set three random variables \begin{equation*} X_1, X_2, X_3 \sim \mathcal{N}(\mu, \sigma) \end{equation*} and their respective transformations \begin{equation*} Y_1 = |X_1| \;,\; Y_2=|X_2| \;,\; Y_3= |X_3|. \end{equation*} I know the densities for…
Simon
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Is there a analytical formula for Super- and Sub-Poissonian distributions?

I'm currently wrtiting my Bachelors thesis on photon statistics. The way different sources of light can be classified is by Poissonian (coherent light), Super-Poissonian (thermal light) and Sub-Poissonian (n-Photon-states). However, both books that…
John Doe
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what is the intuition behind Delta method?

I'm trying to learn the delta method in probability but couldn't quite get the hang of it. For example: trying to solve a problem from the book statistical Inference : Consider a random sample from $\mathrm{Beta}(\alpha ,\beta)$ distribution $\alpha…
0x0
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Inverse distribution function

Let $F$ be a distribution function and $X$ a random variable which is uniformly distributed on the interval $(0,1)$. Let $F^{-1}$ be the inverse defined by $F^{-1}(y)=\inf\{x:F(x)\geq y\}$. How do I show that the random variable $Y=F^{-1}(X)$ has…
JimmyP
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What are some Poisson-like distributions over a finite range of integers?

I'm writing a program in which, in any given time step, a random number of messages is sent. The number of messages is always between $0$ and $n$. I want to be able to control the probability, so that I can cause more or fewer messages to be most…
Mars
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Reliability function, proving exponential distribution

We are given $R(t)$ = $P(X>t)$ for all $x > 0$ and $$R(0) = 1 - Fx(0) = 1\text{ and }\lim\limits_{t \to \infty} R(t) = 0$$ The random variable $X$ also satisfies the memoryless property: $$P(X>s+t|X>t) = P(X>s)\text{ for }s>0\text{ and }t>0$$ Let…
Alistair
  • 271
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What is the distribution of $x'Cx$ when $x$ is a standard gaussian vector

When $x$ is an '$n$' dimensional standard Gaussian, we have $x'x \sim \chi^2$ with $n$ degrees of freedom. Now if I have a symmetric matrix $C$, what will be the distribution of $x'Cx$ ? $C$ is the inverse of a positive definite matrix, like an…
NSR
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Projection of a uniform distribution on a sphere

Suppose $X = (X_1, X_2, X_3)$ is a random vector distributed uniformly on the unit sphere in $\mathbb{R}^3$. What is the probability density function of $X_1$?
user66081
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Score necessary to win a dice rolling tournament

Let's say we have a tournament with 80000 people in it, where each rolls a 10 sided dice 9 times. The highest sum of the 9 dice rolls wins the tournament. How do we calculate the average sum of dice rolls that will win the tournament? Assume the…
Daniel Steinberg
  • 31
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Assistance please on distribution problems

How many ordered quadruples (a,b,c,d) satisfy a+b+c+d=18, where a,b,c,d are odd positive integers? How many ordered quadruples (a,b,c,d) satisfy a+b+c+d=18, where a,b,c,d are integers such that |a|,|b|,|c|,|d| are each at most 10? For some reason, I…
Mathy Person
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Compute the mean and variance given a probability mass function

I'm given the formula: $(1-p)^{x-1}p, x = 1, 2, ..., \infty$ and I'm asked to find the mean and variance. I know the mean is represented by $\sum_{i=1}^n p_ix_i$ and the variance by $\sum_{i=1}^n p_i(x_i-\mu)^2$, but I'm not really sure how to get…
Ryan
  • 1,651
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Exact and approximate probability distribution

Here is a question that I want to understand how to solve it. At a university, a lecture hall holds $n = 400$ students. Realizing that some students invariably drop a class, the registrar overbooks the lecture hall for a particular class,…
erkangur
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Expectation of $\frac{1-\Phi(\frac{Y-\mu_z}{\sigma_z}-\sigma_z)}{1-\Phi(\frac{Y-\mu_z}{\sigma_z})}$ of normal random variable.

The question is: Suppose that X and Y are independent Normally distributed random variables, $$X\sim N(a,\sigma_1^2)$$ $$Y\sim N(b,\sigma_2^2)$$ and $Z = \rho X + \sqrt{(1-\rho^2)}Y$, find $E[max(0,e^Z-e^Y)]$. My solution is as follows: $(Z,Y)$ has…
Tianyi Jia
  • 281
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Mulivariate Probability Distribution

Let X and Y have the joint probability density function given by: f(x,y)=$\frac{1}{4}exp{\frac{-(x+y)}{2}}$, x>0 and y>0 (a) Find Pr(x<1,y>1) (b) Find Pr(y<$x^2$) This is how I tackled (a): Pr(x<1,y>1)=$\int\int exp{\frac{-(x+y)}{2}}$ dydx where…
J.R.
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